Quotient group

A quotient group is a group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure.

Definitions

Let $G$ be a group and $N⊴G$ a normal subgroup.

Definition via cosets

The quotient group $G/N$ is the set of left cosets $$G/N:=\{gN\mid g\in G\}$$with group operation $(gN)(hN):=(gh)N.$

Definition via equivalence relations

Define a relation on $G$ by$$g\sim h⟺g^{-1}h\in N.$$

The relation $\sim$ is a equivalence relation, and denote $[g]:=\{a\mid a\sim g\}$ as the equivalence class of $g$. The quotient group is defined by the set of equivalence classes$$G/N:=G/\sim$$with operation $[g][h]:=[gh].$